The cartesian fibration \(\mathcal {U} \rightarrow \mathbbe {\Delta }\)  determines a right fibration \(\mathcal {U}^{\operatorname{cart}} \rightarrow \mathbbe {\Delta }\) , and hence by straightening ([1]}, Theorem. 8.3.1) a simplicial groupoid
\(U: \mathbbe {\Delta }^{\operatorname{op}} \rightarrow \widehat{\mathbf {Grpd}}\) 
